More on Seiberg-Witten theory and monstrous moonshine: a new simple method of calculating the prepotential. (English) Zbl 1521.81107
Summary: We continue the study of a relationship between the instanton expansion of the Seiberg-Witten (SW) prepotential of \(D = 4\), \(\mathcal{N} = 2\) \(SU(2)\) SUSY gauge theory and the Monstrous moonshine. As was done in [S. Mizoguchi, PTEP, Prog. Theor. Exper. Phys. 2022, No. 12, Article ID 121B01, 9 p. (2022; Zbl 1519.81397)], we expand the inverse of the modular \(j\)-function in \(u^{-1}\) around \(u = \infty\), where \(u\) is the familiar \(u\) parameter for the respective SW curves, and compute the complex gauge coupling \(\tau\) as a function of the scalar vev \(a\) by using the Fourier expansion of \(j(\tau)\) and the relation between \(u\) and \(a\) obtained by the Picard-Fuchs equation. In this way, the instanton expansion of the prepotential is related to the dimensions of representations of the Monster group. We show that, for the cases of \(N_f = 2\) and 3, \(q\) again has an expansion whose coefficients are all integer-coefficient polynomials of the moonshine coefficients if the expansion variables are appropriately chosen. This hints some unknown relation between the Liouville CFT and the vertex operator algebra CFT with different central charges. We also demonstrate that this new method of calculating the SW prepotential developed here is useful by performing some explicit computations.
MSC:
| 81R12 | Groups and algebras in quantum theory and relations with integrable systems |
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 41A58 | Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) |
| 14H15 | Families, moduli of curves (analytic) |
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
Keywords:
Picard-Fuchs equationCitations:
Zbl 1519.81397References:
| [1] | Seiberg, N.; Witten, E., Nucl. Phys. B, 426, 19 (1994); Seiberg, N.; Witten, E., Erratum, Nucl. Phys. B, 430, 485 (1994) |
| [2] | Seiberg, N.; Witten, E., Nucl. Phys. B, 431, 484 (1994) · Zbl 1020.81911 |
| [3] | J. McKay, In a letter to John G. Thompson, 1978, unpublished. |
| [4] | Thompson, J. G., Bull. Lond. Math. Soc., 11, 352-353 (1979) · Zbl 0425.20016 |
| [5] | Conway, J. H.; Norton, S. P., Bull. Lond. Math. Soc., 11, 308-339 (1979) · Zbl 0424.20010 |
| [6] | Frenkel, I. B.; Lepowsky, J.; Meurman, A., Proc. Natl. Acad. Sci. USA, 81, Article 32566 pp. (1984) |
| [7] | Frenkel, I. B.; Lepowsky, J.; Meurman, A., (Vertex Operators in Mathematics and Physics (1985), Springer-Verlag: Springer-Verlag New York), 231 |
| [8] | Dixon, L. J.; Ginsparg, P. H.; Harvey, J. A., Commun. Math. Phys., 119, 221-241 (1988) · Zbl 0657.17011 |
| [9] | Borcherds, R., Invent. Math., 109, 405-444 (1992) · Zbl 0799.17014 |
| [10] | Mizoguchi, S., PTEP, 2022, 12, Article 121B01 pp. (2022) · Zbl 1519.81397 |
| [11] | Tachikawa, Y., Lect. Notes Phys., 890 (2014) |
| [12] | McKay, J.; He, Y. H., ICCM Not., 10, 1, 71-88 (2022) · Zbl 1511.20061 |
| [13] | Malmendier, A.; Ono, K. |
| [14] | Moore, G. W.; Witten, E., Adv. Theor. Math. Phys., 1, 298-387 (1997) · Zbl 0899.57021 |
| [15] | Korpas, G.; Manschot, J.; Moore, G. W.; Nidaiev, I., Res. Math. Sci., 8, 43 (2021) · Zbl 1472.81184 |
| [16] | Eguchi, T.; Ooguri, H.; Tachikawa, Y., Exp. Math., 20, 91-96 (2011) · Zbl 1266.58008 |
| [17] | Witten, E., Commun. Math. Phys., 117, 353 (1988) · Zbl 0656.53078 |
| [18] | Nekrasov, N. A., Adv. Theor. Math. Phys., 7, 5, 831-864 (2003) · Zbl 1056.81068 |
| [19] | Nekrasov, N.; Okounkov, A., Prog. Math., 244, 525-596 (2006) · Zbl 1233.14029 |
| [20] | Klemm, A.; Lerche, W.; Theisen, S., Int. J. Mod. Phys. A, 11, 1929-1974 (1996) · Zbl 1044.81739 |
| [21] | Masuda, T.; Suzuki, H., Int. J. Mod. Phys. A, 12, 3413-3431 (1997) · Zbl 1101.14301 |
| [22] | Ohta, Y., J. Math. Phys., 37, 6074-6085 (1996) · Zbl 0863.58080 |
| [23] | Ohta, Y., J. Math. Phys., 38, 682-696 (1997) · Zbl 0918.58078 |
| [24] | Chan, G.; D’Hoker, E., Nucl. Phys. B, 564, 503-516 (2000) · Zbl 0956.81084 |
| [25] | Dijkgraaf, R.; Vafa, C., Toda theories, matrix models, topological strings, and N=2 gauge systems |
| [26] | Eguchi, T.; Maruyoshi, K., J. High Energy Phys., 1007, Article 081 pp. (2010) |
| [27] | Itoyama, H.; Oota, T.; Yano, K., Phys. Lett. B, 789, 605-609 (2019) · Zbl 1406.81061 |
| [28] | Mizoguchi, S.; Otsuka, H.; Tashiro, H., Phys. Lett. B, 800, Article 135075 pp. (2020) · Zbl 1434.81069 |
| [29] | Closset, C.; Magureanu, H., SciPost Phys., 12, 2, Article 065 pp. (2022) |
| [30] | Huang, M.x.; Kashani-Poor, A. K.; Klemm, A., Ann. Henri Poincaré, 14, 425-497 (2013) · Zbl 1272.81119 |
| [31] | Alday, L. F.; Gaiotto, D.; Tachikawa, Y., Lett. Math. Phys., 91, 167 (2010) · Zbl 1185.81111 |
| [32] | Nahm, W. |
| [33] | Aspman, J.; Furrer, E.; Manschot, J., Phys. Rev. D, 105, 2, Article 025021 pp. (2022) |
| [34] | Magureanu, H., J. High Energy Phys., 05, Article 163 pp. (2022) |
| [35] | Aspman, J.; Furrer, E.; Manschot, J., Ann. Henri Poincaré, 22, 8, 2775-2830 (2021) · Zbl 1472.81160 |
| [36] | T. Oikawa, in preparation. |
| [37] | Cheng, M. C.N.; Duncan, J. F.R.; Harvey, J. A., Commun. Number Theory Phys., 08, 101-242 (2014) · Zbl 1334.11029 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
